w10
plain-language theorem explainer
The raw flat-angle Jacobian coefficient at edge-slot pair (1,0) is exactly zero. Zero-mode free-coefficient identities and full-star homothety stationarity proofs cite this table entry when expanding the six-type stencil sum. Proof is a one-line rewrite through the evaluated rational stencil table, then numeric normalization.
Claim. The single-entry radical coefficient of the flat-angle Jacobian at tetrahedron edge indices $f=1$, $g=0$ equals zero: $J_{1,0}/(2\sqrt{a^*_1})=0$ in $\mathbb{R}$.
background
This module sits in the QG full-theory campaign (Paper C / Pillar 1, Lane C) and closes the hinge-aware zero-mode gate for the assembled Regge TT constant block. The raw coefficient is the radical-normalized single entry $J_{fg}/(2\sqrt{a^*_f})$ built from the flat-angle Jacobian and Freudenthal squared edge lengths; it is not a fiber sum.
Upstream, every one of the 36 raw coefficients is identified with a literal rational stencil weight via rawJacobianCoefficient_eval: the independent phase-free match table (all slot pairs) records the closed-form value of that radical expression. For the ordered pair $(1,0)$ the table entry is the rational $0$.
The local setting contrasts stencil-only residuals (nonzero at the TT witness) with the assembled hinge-minus-stencil block, which cancels. Individual vanishing entries such as this one are the arithmetic atoms of that assembled contraction.
proof idea
One-line wrapper. Rewrite the left-hand side by the universal evaluation theorem that equates every raw coefficient to the corresponding rational stencil weight cast to $\mathbb{R}$. Then norm_num on the full literal rational table discharges the concrete match clause for left index 1 and right index 0, which is the constant $0$.
why it matters
The entry is consumed by zeroMode_free_coefficients, which expands the six-tetrahedron raw-table contraction in seven free edge-class coefficients and obtains the perfect square $(c_0+c_1+c_2-c_3-c_4-c_5+c_6)^2/2$. That algebraic shape is the exact form of the assembled constant block and vanishes on the alternating-sum hyperplane where geometric edge-class vectors live.
It is also pulled into the three full-star homothety stationarity theorems (fullStar_homothety_stationary in the 4D star-kernel variants), which show the directional full-star sum against natural class weights is zero. Within Gate C-A3 this is bookkeeping that makes the assembled $k=0$ block computable without residual radicals, supporting the headline that the Schlaefli-reduced assembled constant block vanishes at the TT witness.
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